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polyTop: Software for Computing Topology of Smooth Real Surfaces

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Mathematical Software – ICMS 2018 (ICMS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10931))

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Abstract

A common computational problem is to compute topological information about a real surface defined by a system of polynomial equations. Our software, called polyTop, leverages numerical algebraic geometry computations from Bertini and Bertini_real with topological computations in javaPlex to compute the Euler characteristic, genus, Betti numbers, and generators of the fundamental group of a smooth real surface. Several examples are used to demonstrate this new software.

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Notes

  1. 1.

    Available at http://dx.doi.org/10.7274/R0PV6HF4.

References

  1. Bates, D.J., Brake, D.A., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: On computing a cell decomposition of a real surface containing infinitely many singularities. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 246–252. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44199-2_39

    Chapter  Google Scholar 

  2. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for numerical algebraic geometry. bertini.nd.edu

  3. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Numerically solving polynomial systems with Bertini. In: SIAM (2013)

    Google Scholar 

  4. Berger, M., Tagliasacchi, A., Seversky, L.M., Alliez, P., Levine, J.A., Sharf, A., Silva, C.T.: State of the art in surface reconstruction from point clouds. In: Eurographics 2014 - State of the Art Reports. The Eurographics Association (2014)

    Google Scholar 

  5. Besana, G.M., Di Rocco, S., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Cell decomposition of almost smooth real algebraic surfaces. Num. Alg. 63(4), 645–678 (2013)

    Article  MathSciNet  Google Scholar 

  6. Brake, D.A., Bates, D.J., Hao, W., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Algorithm 976: Bertini\_real: numerical decomposition of real algebraic curves and surfaces. ACM Trans. Math. Softw. 44(1), 10 (2017). bertinireal.com

    Article  MathSciNet  Google Scholar 

  7. Brake, D.A., Bates, D.J., Hao, W., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini\_real: software for one- and two-dimensional real algebraic sets. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 175–182. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44199-2_29

    Chapter  MATH  Google Scholar 

  8. do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, New Jersey (1976)

    Google Scholar 

  9. Cucker, F., Krick, T., Shub, M.: Computing the homology of real projective sets. Found. Comput. Math. 15, 281–312 (2015)

    Google Scholar 

  10. Dufresne, E., Edwards, P.B., Harrington, H.A., Hauenstein, J.D.: Sampling real algebraic varieties for topological data analysis. arXiv:1802.07716 (2018)

  11. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  12. Munkres, J.R.: Topology. Prentice Hall, New Jersey (2000)

    Google Scholar 

  13. Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Disc. & Comput. Geom. 389(1–3), 419–441 (2008)

    Article  MathSciNet  Google Scholar 

  14. Oudot, S., Rineau, L., Yvinec, M.: Meshing volumes bounded by smooth surfaces. In: Hanks, B.W. (eds.) Proceedings of the 14th International Meshing Roundtable, Sandia National Laboratories. Springer, Heidelberg, pp. 203–220 (2005). https://doi.org/10.1007/3-540-29090-7_12

  15. Adams, H., Tausz, A., Vejdemo-Johansson, M.: javaPlex: a research software package for persistent (co)homology. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 129–136. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44199-2_23

    Chapter  MATH  Google Scholar 

  16. Tretkoff, C.L., Tretkoff, M.D.: Combinatorial group theory, Riemann surfaces and differential equations. Contemp. Math.: Contrib. Group Theory 33, 467–519 (1984)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors thank Mikael Vejdemo-Johansson for input regarding javaPlex. All authors acknowledge support from NSF ACI-1440607/1460032. Additional support for JDH was provided by Sloan Research Fellowship BR2014-110 TR14 and for MHR by Schmitt Leadership Fellowship in Science and Engineering.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire, USA

    Danielle A. Brake

  2. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, USA

    Jonathan D. Hauenstein & Margaret H. Regan

Authors
  1. Danielle A. Brake
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  2. Jonathan D. Hauenstein
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  3. Margaret H. Regan
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Corresponding author

Correspondence to Margaret H. Regan .

Editor information

Editors and Affiliations

  1. University of Bath, Bath, United Kingdom

    James H. Davenport

  2. Johannes Kepler University, Linz, Austria

    Manuel Kauers

  3. University of Waterloo, Waterloo, Ontario, Canada

    George Labahn

  4. Czech Technical University in Prague, Prague 6, Czech Republic

    Josef Urban

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Brake, D.A., Hauenstein, J.D., Regan, M.H. (2018). polyTop: Software for Computing Topology of Smooth Real Surfaces. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_47

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  • DOI: https://doi.org/10.1007/978-3-319-96418-8_47

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-96417-1

  • Online ISBN: 978-3-319-96418-8

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